Select the Ocular object then Attach the Cornea object.
Now there is only the Ocular object.
As this Ocular object belongs to the head we will assign the same Multi-Sub/Object material to it.
For that add 2 slots with the Add button and name the new Checker IDs.

Select the Eye and apply this material to it.

Select the internal sphere in Element mode and enter 4 within Material ID, start again for the external sphere with a value of 5.

For more precision on all that, to see the bases of the mapping.

Before placing the co-ordinates of mapping, select the faces shown and delete them.
If we could see it we would need the full sphere. This not being the case we can safely delete the back half.

Apply a Planar Map in face mode.

Adjust the mapping Gizmo with View Align and FIT.

With Texporter, enter 512x512 for the size of the texture, then capture Only ID 4 to recover the UV of the eye and Only ID 5 for the Cornea

On left the UV of the Cornea and on the right those of the Eye.
Now goto Photoshop and begin textures according to these guides.

In Photoshop, place the Guide image of UV on a Layer and apply negative or invert color, using Multiply mode will give us the transparent white.

Add copy layers for each part of the eye texture, to facilitate the final improvements.

Make the reflections in the same way for the texture of Outer Eye, but this time use the Guide image in Screen mode to have the transparent black.

Save the textures in TGA, TIFF or PNG format and KEEP the Original PSD files with all the copies.
Checker ID 4 corresponds with the eye, place this in the Diffuse channel of the Ocular.

And in Checker ID 5, set the colors to white as opposite, boost the specular and place the texture bitmap in the Reflection channel this time, with a value of 20% or less. Take care to activate the mapping by UV, by default it is Environ which is active.
This way makes it possible to add reflections on the cornea without having recalculate an exact calculation of the raytrace environment...

Adjust the orientation of the eyes (the second eye is a simple copy not a mirror of the first).

Click to Enlarge

Click to Enlarge

Some renders with basic scanline.
You can go back and customize the shaders of the eye to give a more personal look to your materials.